Problem: Factor the following expression: $-2$ $x^2+$ $19$ $x$ $-45$
Answer: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-2)}{(-45)} &=& 90 \\ {a} + {b} &=& & & {19} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $90$ and add them together. The factors that add up to ${19}$ will be your ${a}$ and ${b}$ When ${a}$ is ${9}$ and ${b}$ is ${10}$ $ \begin{eqnarray} {ab} &=& ({9})({10}) &=& 90 \\ {a} + {b} &=& {9} + {10} &=& 19 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-2}x^2 +{9}x +{10}x {-45} $ Group the terms so that there is a common factor in each group: $ ({-2}x^2 +{9}x) + ({10}x {-45}) $ Factor out the common factors: $ x(-2x + 9) - 5(-2x + 9) $ Notice how $(-2x + 9)$ has become a common factor. Factor this out to find the answer. $(-2x + 9)(x - 5)$